Yes, I understand that, but how do you know its value?
The answer I get from looking at the reference you gave is this: we know the actual spindown rate of the pulsar from measurements of its pulse period and how that changes over time. So we know the actual rate of rotational energy loss. The question is, how much of that energy loss is due to GWs, and how much is due to EM radiation (which is basically the only other energy loss mechanism available)?
To know that, we have to know the actual EM dipole moment and gravitational quadrupole moment of the pulsar. However, we have no independent way of measuring either of those things. The best we can do is to use the intensity of the EM radiation we detect from the pulsar (namely, its pulses) to estimate how much energy it is radiating as EM waves, and to try to detect GWs from it to get an estimate how much energy it is radiating as GWs.
As I understand it, the EM intensity we see is consistent with all of the observed rotational energy loss being due to EM wave emission. This would indicate that the pulsar is not radiating GWs to any significant extent, and therefore has a negligible gravitational quadrupole moment (or more precisely a negligible third time derivative of same). This is also consistent with the fact that we have observed no GWs from the pulsars in question.
How does all this relate to the quadrupole formula? Well, the quadrupole formula predicts that a negligible third time derivative of the quadrupole moment gives a negligible energy loss due to GW emission. That's why we deduce a negligible third time derivative of the quadrupole moment based on the fact that all of our observations indicate a negligible energy loss due to GW emission. I assume that ##\epsilon## is basically equivalent to "third time derivative of the quadrupole moment" (the latter is what I'm used to seeing in GW emission power formulas), so these observations would also predict a negligible ##\epsilon##.
This doesn't seem to be stated correctly (as mfb has pointed out several times now). The difference between the spindown limit prediction for pulsar GW emission, and the actual observed pulsar GW emission, is in the value of ##\epsilon## we plug into the formula; the formula itself is the same in both cases. For the spindown limit, we use the observed rate of energy loss to infer what the value of ##\epsilon## would have to be to account for all of that energy loss being due to GWs. To match the actual observations, we calculate what the upper limit on ##\epsilon## would be such that the energy loss rate due to the GW quadrupole formula is less than the smallest value we could have measured, i.e., the upper limit on ##\epsilon## consistent with no GWs actually being observed. But whatever that rate is, it is exactly the rate given by the quadrupole formula for that value of ##\epsilon##. There isn't a different formula for the spindown limit case vs. the actual case.